Basically, are my corrupted files going to transfer over to my new harddrive?

Basically, we can't tell.

It depends on what is corrupted (the file data or the FAT), how you are doing the transfer, why the corruption happened in the first place.

It is perfectly possible under the right circumstances for a corrupted file to transfer quite happily - but it will still be corrupted when it gets to the new drive.

Oops. I guess I should have given some background info. My harddrive became corrupted randomly one day when I went to turn on my xbox system. At first, the xbox would not load the dashboard, but after a reboot, I got to a screen that said my harddrive was corrupted. Fortunately, I was able to upload a majority of my save files over to cloud storage, so I could (hopefully) download them later.

I was planning on transferring between drives using the transfer cable made by Microsoft. Also, my old harddrive is the 20gb and I plan on getting the 250gb harddrive.

You should also have posted in the right section - this is Xbox 1. Hopefully a mod will move it, and someone with a deep working knowledge of Xbox 360 hard drive corruption and data recovery will be able to help.